Zandile borrowed money from a bank and she is repaying the loan by making monthly payments of R1262 at the end of every month, each payment includes interest and repayment of the principle. The loan will be paid off in 9 years and interest is calculated at a rate of 12.742% per year, compounded monthly. What is the principle repayment component of the 6th payment of the 6th year? (Give your answer rounded to 2 decimal places. If you think the answer is R1012.3456, type in 1012.35)

To solve this problem, we can use the formula for the balance of a loan paid with equal monthly installments. The principle repayment component of the 6th payment in the 6th year is determined by the remaining balance after the 5th payment of the 6th year, as well as the interest paid on the remaining balance.

### Step-by-step approach:

1. **Loan Amount (P):**  
   We need to first find the original loan amount using the monthly payment formula:

   \[
   P = \frac{M \times \left(1 - (1 + r)^{-n}\right)}{r}
   \]

   where:
   - \( M = 1262 \) (the monthly payment),
   - \( r = \frac{12.742\%}{12 \times 100} = 0.01061833 \) (monthly interest rate),
   - \( n = 9 \times 12 = 108 \) (total number of payments).

2. **Remaining Balance after the 5th Year:**
   To find the principal repayment in the 6th payment of the 6th year, we first need to calculate how much of the loan is still outstanding after 5 years (or 60 payments).

   The balance of the loan after 60 payments can be computed using the following formula:

   \[
   B_k = P \times (1 + r)^k - \frac{M \times \left((1 + r)^k - 1\right)}{r}
   \]

   where \( k = 60 \) represents the 60th payment.

3. **Interest for the 6th Payment of the 6th Year:**
   Once we have the remaining balance after 5 years, the interest portion of the 6th payment of the 6th year is calculated as:

   \[
   \text{Interest portion} = B_k \times r
   \]

4. **Principal Repayment for the 6th Payment:**
   The principal repayment portion of the 6th payment is:

   \[
   \text{Principal repayment} = M - \text{Interest portion}
   \]

I'll calculate this step-by-step now.The principal repayment component of the 6th payment of the 6th year is R760.11.

Would you like any further details or have any questions about the solution? 

Here are 5 related questions to expand on this topic:
1. How would the loan terms change if the interest was compounded quarterly instead of monthly?
2. What is the total amount of interest paid over the full loan term?
3. How much of the loan balance remains after 6 years of payments?
4. How would the monthly payment change if the interest rate was reduced to 10% per annum?
5. How much of the 7th payment in the 6th year goes towards the principal?

**Tip:** When repaying loans, more of the monthly payment goes toward interest at the start, but this shifts toward paying off more of the principal over time.

Learn how to calculate the principal repayment component of the 6th payment in the 6th year of a loan with monthly payments of R1262, an interest rate of 12.742% per year compounded monthly, and a repayment term of 9 years. This detailed solution involves loan amortization, compound interest, and principal repayment formulas. Suitable for advanced high school and college-level students.

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